Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter

We examine the equation $${\mathrm{\Delta }}^{2}u=\lambda f(u)\mathrm{\Omega },$$ with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $\lambda$. We obtain similar results for the sytem {equation*} {{array}{rrl} -\Delta u &=& \lambda f(v) \qquad \Omega, -\Delta v &=& \gamma g(u) \qquad \Omega, u&=& v = 0 \qquad \partial Omega. {array}. {equation*}